Algebraic Topology

Math 427/527

Winter 2021

This course covers some essential elements of algebraic topology. It has, very broadly, three parts:

Part 1: Homology

Part 2: Cohomology

Part 3: Selected topics drawing on duality, infinite cyclic covers, characteristic classes

Everything you need for the course is here, apart from zoom links and homework submission, which makes use of Canvas. To see what this course looked like the last time I taught it, go here.

Please note that Math 426 (or similar) is an essential prerequisite. In particular, familiarity with the fundamental group will be assumed.

This term is starting with some unknowns (for the latest, go here), and it seems likely that flexibility will be required. I will use your official UBC email to contact you where needed. Ensure that your official email is up-to-date (namely, that you actually receive/read emails from your official email) at the Student Service Centre.

The image on the right was borrowed from The Inverse Homotopy by Tom Hockenhull (with permission); read the entire comic strip here.

• Instructor
  Liam Watson
  liam(at)math(dot)ubc(dot)ca
  Office hours by appointment.

• Where, when...
  Starts online, using zoom; from February 7 we meet in in Math Annex 1102. Class is on Wednesdays and Fridays from 830 to 945.

• Homework
  There will be at least 4 homework assignments – at this stage I think it makes sense to remain flexible due to the evolving situation associated with Covid. You will always have at least one full week (and usually more) to complete assignments; deadlines and assignments will be posted here. Your assignments should be prepared using LaTex. Please note that no late assignments will be accepted.
• Homework 1 was posted on January 31 and is due on Friday February 11 by 5:00 pm.
• Homework 2 was posted on February 17 and is due on Wednesday March 2 by 5:00 pm.
• Homework 3 was posted on March 16 and is due on Wednesday March 30 by 5:00 pm.
• Homework 4 was posted on April 1 and is due on Wednesday April 13 by 5:00 pm.

• Evaluation
  This course is evaluated entirely on homework. Each assignment carries equal weight.

• Suggested references
  Algebraic Topology by Allen Hatcher, available here.
  Basic Category Theory by Tom Leinster, available here.
  
  A word about supporting material for the course: The references listed above have been designated as optional, but this does not mean that seeking supporting material for the course is not required. There are many great references for topology, and it is up to you to find the materials you need to complement the lectures and succeed in the course. This search may be done in consultation with me; I am more than happy to help. I will endeavour to provide clear notes in class, summarise lectures below, and point to references for additional reading.

• Further reading
  Frobenius Algebras and 2D Topological Quantum Field Theories by Joachim Kock, available online through the UBC library.
  Topology from the Differentiable Viewpoint by John Milnor available at the UBC library.
  Morse Theory by John Milnor available at the UBC library.

              Lecture summaries

• Lecture 1: Introduction (January 12)
  I like this quote from Tom Leinster: "A category is a system of related objects. The objects do not live in isolation: there is some notion of map between objects, binding them together." Today we reviewed some basics of category theory, particularly functors between categories, and used this to formailze a proof of the Brower fixed point theorem.
• Lecture 2: Euler characteristic (January 14)
  Our aim is to compute a suite of functors called homology groups. These will be relatively easy to work with, but the cost is a somewhat lengthy definition. To motivate the objects we'll require, we looked at the Euler characteristic of a surface, which suggests a notion of cellular decomposition (to be formalized). Our goal, ultimately, is to find a categorical lift of the Euler characteristic.
• Lecture 3: CW complexes (January 19)
  In order to define homology groups for X, we need some sort of decomposition of the space X. There are various ways to do this. To start with, we will use CW complexes, which were introduced today.
• Lecture 4: Some intuition for homology (January 21)
  By considering low-dimensional CW complexes, namely graphs, we attempted to generate some intuition for what homology might measure. Ultimately, we concluded that the one dimensional homology of a space should count cycles (essentially un-based loops) modulo boundaries. That is, while the 1-dimensional chain group is generated by the 1-cells, the 1-dimensional homology classes were equivalence classes of those cycles differing by some boundary.
• Lecture 5: Homology as a functor (January 26)
  We collected our intuition from last lecture into a suite of definitions, including chain complexes of modules and the homology of such complexes. This allowed us to axiomatically define the functors that we wish to construct (and single out their essential properties), taking the based homotopy category to a category of modules. The image of these functors, which we still need to construct, will ultimately be the homology modules (groups, when the underlying ring is Z) associated with a CW complex.
• Lecture 6: Homology is stable (January 28)
  Continuing our discusion from last time, we showend that homology is stable with respect to suspension. From this fact it follows that a homotopy equivalence of spheres S^m and Sⁿ is equivalent to m=n. While this still depends on a class of functors that we have yet to fully construct, it does indicate that the attaching maps we are interested in leveraging into chain maps should be completely determined by their degree, that is, the value of the identity in the ring R in the image of a morphism induced from a map between spheres.
• Lecture 7: The degree of a map between spheres (February 2)
  We formalized the notion of degree, and checked the basic properties forced on us by functoriality.
• Lecture 8: The local degree (February 4)
  We'd like to be able to calculate the degree by hand, in fact, what we would really like is for all of this to line up with the attaching maps coming from our cell complexes. To this end, the right definition of local degree follows from removing a point y of Sⁿ in the image of f and removing the preimage of this point x₁,...,x_m in Sⁿ. The fact that this was the right thing to do followed from a discussion about the Eilenberg-Steenrod axioms (in relation to our definition of reduced homology groups).
• Lecture 9: Degree as a sum of local degrees (February 9)
  We gave a definition of the degree of a map in terms of the local degree, as a generalization of winding number.
• Lecture 10: Degree à la Milnor (February 11)
  A long aside following Milnor about how we might compute local degree in terms of regular values associated with a smooth map of spheres. For those of you with background/inrerest in geometry, Milnor's book is a great short read.
• Lecture 11: Cellular homology (February 16)
  Finally, we assembled all the pieces and defined the cellular chain complex. This came with a proof of a theorem stated much earlier: for any ring R the reduced homology functors we gave as a definition were completely determined by the axioms in said definition.
• Lecture 12: Cell complexes via Morse theory (February 18)
  Expanding on the discussion started last time, we saw in detail how a Morse-Smale function on smooth, compact manifold generates a cell structure. Indeed, these considerations allowed us to define the Morse-Witten chain complex, and even see parts of the proof that this complex computes the cellular homology groups we've been studying.

• Reading week
  Please read section 2.1 of Hatcher on Simplicial and singular homology (pages 102 through 131). Important points are homotopy invariance (which you should understand the details of) and excision (which you should at least be aware of).

• Lecture 13: Introduction to cohomology, by example (March 2)
  We are now shifting gears from homology to cohomology. In order to motivate why this might be an interesting thing to to, today we thought trough an essay by Roger Penrose on the cohomology of impossible figures which describes how to "measure" the impossibility (that is, the fact that as a 2-dimensional drawing it cannot represent a 3-dimensional object) of the tribar as a cohomology class.
• Lecture 14: The Ext functor (March 4)
  Given and chain complex of abelian groups, we define a cochain complex by dualizing with Hom(-,G). We looked in detail at the way in wich this functor determines the cohomology from the homology: it does, up to a functor Ext(-,G).
• Lecture 15: Tensor products and the Tor functor (March 9)
  Guest lecture by Ben Williams.
• Lecture 16: The Künneth formula (March 11)
  Guest lecture by Ben Williams.
• Lecture 17: The cup product (March 16)
  We built a product in cohomology using the dualized diagonal map. As a result, the cohomology groups H*(X) may be regarded as a graded ring. Our approach is meant to be complementary to Hatcher's Section 3.2, which is assigned reading as part of Homework 3.
• Lecture 18: Orientations on manifolds (March 18)
  We used the infinite cyclic local homology H_n(M|x) of an n dimensional manifold M to give a notion of orientability inheriting familiar properties that we are used to from Rⁿ. Of particular use going forward will be the fundamental class of an orientable manifold, which is an element of H_n(M) whose image in H_n(M|x) is a generator, for all points x in M.
  
  Relevant reading: Hatcher's subsection Orientations and Homology, pages 233 through 239.
• Lecture 19: The cap product (March 23)
  The cap product uses the evaluation map together with the diagonal map in order to pair a cohomology class with a homology class. In particular, when M is an orientable manifold and [M] is a fundamental class, then capping with [M] gives rise to the isomorphism in Poincaré duality.
  
  Relevant reading: first part of Hatcher's subsection The Duality Theorem, pages 239 through 242, together with Hatcher's subsection Cohomology of Spaces, pages 197 through 204.
• Lecture 20: Cohomology with compact support and duality (March 25)
  Using directed systems of cohomology groups associated with compact subspaces we defined homology with compact support via limits of abelian groups. This is a key to the proof of Poincaré duality, but also gives rise to other duality theorems such as Alexander duality. This latter highlights some properties that have come up earlier in the course when we were studying infinite cyclic covers and computing the modules associated with them.
  
  Relevant reading: Hatcher's subsection The Duality Theorem, pages 239 through 249, together with the subsections that follow: Connections with Cup Product and Other forms of Duality.
• Lecture 21: Infinite cyclic covers (March 30)
  Today focused on a concrete interplay between algebra and topology: we considered CW complexes equipped with a cover having infinite cyclic deck group and concluded that the homology (with field coefficients) of the cover carried a natural and explicit module structure over a ring of Laurent polynomials. This latter being a PID, a natural invariant of the cover is the order of the module in each homological degree, which in this case is a product of laurent polynomials.
  Incidentally, as part of your homework you'll make use of a short exact sequence inducing a long exact sequence: This clip from the film It's my turn gives a really quick proof of the Snake Lemma.
• Lecture 22: Abelianization and knot complements (April 1)
  After sketching the proof the that the Hurewicz map abelianizes the fundamental group of a path connected space and recovers the first homology group, we set about considering a lengthy example by considering the complement of an explicit knot. Our aim, ultimately, is to show that the module structure on the infinite cyclic cover of the knot complement records structure despite the fact that the homology of the complement itself is rather boring. To find a homology generator for our knot, we picked a nice presentation for the fundamental group and argued as in George K. Francis' Topological picturebook; a recreational trip to the Barber stacks is recommended.
• Lecture 23: The Alexander invariant (April 6)
  Today we gave a complete calculation of the Alexander invariant of the trefoil knot (that is, the module structure on the infinite cyclic cover of the knot complement), building on the material developed last time. An important geometric step led us to consider the analog of Seifert-van Kampen for homology: The Mayer-Vietoris exact sequence.
• Lecture 24: Fox free calculus (April 8)
  To round out our discussion on the homology of an infinite cyclic covers, we calculated the differential on the 2-cells explicitly as a map of Z[t,t^-1]-modules. This can be given as a matrix with entries determined by the Fox free derivative applied to the relators in a group presentation for the fundamental group of the base.